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A Bayesian Treasure Hunt

Q: Four friends A,B,C,D learn that a square tract of land they own has a treasure buried somewhere. They divide the tract of land into four equal quadrants, estimate that the probability in each of those square tracts as \(\{r,r,r,p\}\). A starts digging his quadrant first. The probability that A might miss the treasure given the dig is \(q\). Having dug out his quadrant, A exclaims that he still hasn't found the treasure. What is the updated probabilities of

A: This is a classic Bayesian puzzle. The fact that A has dug up his quadrant and revealed that there is no treasure leaves open the possibility that he might have missed the treasure. Yet it is information that must be worth something which will help B,C,D update their belief that the treasure is present in their quadrant. First off lets calculate A's updated belief. A's prior is \(p\). Let \(T\) denote the event that the treasure is in A's quadrant given nothing showed up during the dig. Let \(M\) denote the event that A's dig resulted in a miss. We know from Bayes theorem
$$
P(T|M) = \frac{P(M|T)P(T)}{P(M)}
$$
\(P(M)\) can occur in two ways. Either, the treasure just isn't there or it was there and A missed it. So, \(P(M)\) works out as
$$
P(M) = (1-q)\times p + (1 - p) = 1 - pq
$$
\(P(M|T) = 1 - q\) and \(P(T)=p\). So A's updated belief that the treasure still resides in his quadrant is
$$
P(T|M) = \frac{p(1-q)}{1 - pq}
$$
As far as B is concerned, let \(T_{B}\) denote the event that the treasure is in his quadrant. Casting this in the Bayesian framework yields
$$
P(T_{B}|M) = \frac{P(M|T_{B}) P(T_{B})}{P(M)}
$$
Note, the denominator remains the same. The numerator works out as: \(P(M|T_{B}) = 1\) and \(P(M) = r\). So the update rule for B (and for that matter any other person) is
$$
P(T_{B}|M) = \frac{r}{1 - pq}
$$

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.